Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Use tree diagrams for compound events.
• Use two-way contingency tables for probabilities.
• Distinguish between independent and dependent events.
• Calculate conditional probabilities.
• Solve problems with mutually exclusive events.

Lesson summary

This week, we delve deeper into the fascinating world of probability. Probability is about understanding the likelihood of events occurring. It’s not just about games of chance; it's a fundamental tool used in fields like medicine, finance, weather forecasting, and even everyday decision-making. Consider, for example, understanding the likelihood of load shedding on a particular day, or the chances of winning a bursary. These scenarios, relevant to South African learners, rely on probability concepts. This week focuses on using tree diagrams and two-way contingency tables to solve probability problems, as well as introducing the concepts of independent and dependent events.

Lesson notes

2.1 Tree Diagrams: A tree diagram is a visual tool that helps you organize and calculate probabilities in situations where there are multiple events happening in sequence. Each branch represents a possible outcome, and the probabilities are written along the branches.

How it Works: The diagram starts with a single point, representing the initial event. From that point, branches are drawn representing the possible outcomes of the first event. At the end of each branch, another set of branches can be drawn representing the possible outcomes of the second event, and so on. The probability of each outcome is written along the branch. To find the probability of a sequence of events, you multiply the probabilities along the corresponding branches.

Example: Imagine a coin is flipped twice. What is the probability of getting heads followed by tails?

Step 1: Draw the tree diagram.

The first flip has two possibilities: Heads (H) or Tails (T). The second flip also has two possibilities: Heads (H) or Tails (T).

Step 2: Assign probabilities. P(H) = 1/2 and P(T) = 1/2 for each flip (assuming a fair coin).

Step 3: Calculate the probability of the desired sequence. The path we want is H followed by

T. So, the probability is (1/2) (1/2) = 1/4. 2.2 Two-Way Contingency Tables: A two-way contingency table is a table that summarizes the relationship between two categorical variables. It shows the frequency of each combination of categories.

How it Works: One variable is listed across the rows, and the other is listed across the columns. Each cell in the table shows the number of observations that fall into that combination of categories. You can use the table to calculate probabilities by dividing the number of observations in a particular cell by the total number of observations.

Example: A survey was conducted at a school to see if there was a relationship between taking Maths Literacy and playing sports.

The results are shown below: | | Plays Sports | Doesn't Play Sports | Total | |-------------------|-------------|---------------------|-------| | Maths Literacy | 80 | 40 | 120 | | Not Maths Literacy| 30 | 50 | 80 | | Total | 110 | 90 | 200 | What is the probability that a randomly selected student takes Maths Literacy? P(Maths Literacy) = 120/200 = 3/5 What is the probability that a randomly selected student plays sports? P(Plays Sports) = 110/200 = 11/20 What is the probability that a randomly selected student takes Maths Literacy AND plays sports? P(Maths Literacy and Plays Sports) = 80/200 = 2/5 2.3 Independent and Dependent Events: Independent Events: Two events are independent if the occurrence of one event does not affect the probability of the other event occurring.

Example: Flipping a coin twice. The outcome of the first flip doesn't affect the outcome of the second flip.

Formula: If A and B are independent events, then P(A and B) = P(A) P(B).

Dependent Events: Two events are dependent if the occurrence of one event does affect the probability of the other event occurring.

Example: Drawing two cards from a deck without replacement. The probability of drawing a second card changes depending on what the first card was.

Formula: If A and B are dependent events, then P(A and B) = P(A) P(B|A), where P(B|A) is the probability of B occurring given that A has already occurred (conditional probability). 2.4 Conditional Probability: Conditional probability is the probability of an event occurring given that another event has already occurred. We use the notation P(A|B) to represent the probability of event A occurring given that event B has already occurred.

Formula: P(A|B) = P(A and B) / P(B), where P(B) >

0. Example: Using the survey data from the two-way contingency table above: What is the probability that a student takes Maths Literacy, given that they play sports? P(Maths Literacy | Plays Sports) = P(Maths Literacy and Plays Sports) / P(Plays Sports) P(Maths Literacy | Plays Sports) = (80/200) / (110/200) = 80/110 = 8/11 2.5 Mutually Exclusive Events: Two events are mutually exclusive if they cannot both occur at the same time.

Example: Flipping a coin. You can either get heads or tails, but you can't get both at the same time.

Formula: If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B). Guided Practice (With Solutions)

Question 1: A bag contains 3 red balls and 5 blue balls. A ball is drawn at random, and then without replacing it, another ball is drawn. What is the probability that both balls are red? Use a tree diagram.

Solution: Step 1: Draw the tree diagram.

First draw: Red (R) or Blue (B) Second draw (branches off each of the first draw outcomes): Red (R) or Blue (B)

Step 2: Assign probabilities. P(R on 1st draw) = 3/8 P(B on 1st draw) = 5/8 P(R on 2nd draw | R on 1st draw) = 2/7 (Because one red ball has been removed) P(B on 2nd draw | R on 1st draw) = 5/7 P(R on 2nd draw | B on 1st draw) = 3/7 P(B on 2nd draw | B on 1st draw) = 4/7 Step 3: Calculate the probability of two red balls.